A sharp stability threshold for Erdős Problem 848: admissible sets not contained in a single extremal class have density at most 1/50

Call A ⊆ {1, …, N} admissible if ab+1 is not squarefree for all a, b ∈ A, including a = b. A. C. Li has determined the maximum size f(N) = ⌊(N+18)/25⌋ of an admissible set for every N, in a Lean-formalized proof; the extremal sets are the residue classes n ≡ 7 (mod 25) and n ≡ 18 (mod 25). Sawhney had previously proved a stability statement with an implicit η ≈ 0.0023, leaving open the structure of admissible sets of smaller density. We close that gap in sharp form: for every ε > 0 and all N ≥ N0(ε), every admissible A ⊆ {1, …, N} not contained in a single one of the two classes satisfies |A| ≤ (1/50 + ε)N, and the constant 1/50 is sharp, attained by {n : n ≡ 7, 93 (mod 100)} and {n : n ≡ 43, 57 (mod 100)}. The proof combines uniform analytic density bounds for the relevant squarefree-sieve events with two explicit finite computations: the divisibility (gluing) relation on residues modulo 900, and the maximization of the resulting weighted objective by 0–1 programming together with an exhaustive enumeration over the 439 compatible pairs of Galois-closed signature sets. The argument is self-contained (it uses neither Li's theorem nor Sawhney's) and effective, with an explicit N0(ε) recorded in the paper. This archive contains the paper (LaTeX source and PDF), the stability computations and their outputs, the seating-chart certificates with a checker (verified to N = 9,999,981), and the records of an independent rebuild and kernel replay of Li's Lean proof.